Simplify and expand the following expression: $ \dfrac{2y}{4y + 4}+\dfrac{5y - 1}{y - 1} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(4y + 4)(y - 1)$ Multiply the first term by $\dfrac{y - 1}{y - 1}$ $ \begin{align*} \dfrac{2y}{4y + 4} \times \dfrac{y - 1}{y - 1} & = \dfrac{(2y)(y - 1)}{(4y + 4)(y - 1)} \\ & = \dfrac{2y^2 - 2y}{(4y + 4)(y - 1)}\end{align*} $ Multiply the second term by $\dfrac{4y + 4}{4y + 4}$ $ \begin{align*} \dfrac{5y - 1}{y - 1} \times \dfrac{4y + 4}{4y + 4} & = \dfrac{(5y - 1)(4y + 4)}{(y - 1)(4y + 4)} \\ & = \dfrac{20y^2 + 16y - 4}{(y - 1)(4y + 4)}\end{align*} $ Now we have: $ = \dfrac{2y^2 - 2y}{(4y + 4)(y - 1)} + \dfrac{20y^2 + 16y - 4}{(y - 1)(4y + 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2y^2 - 2y + 20y^2 + 16y - 4}{(4y + 4)(y - 1)} $ $ = \dfrac{22y^2 + 14y - 4}{(4y + 4)(y - 1)}$ Expand the denominator: $ = \dfrac{22y^2 + 14y - 4}{4y^2 - 4}$ Simplify: $ = \dfrac{11y^2 + 7y - 2}{2y^2 - 2}$